In physics, Planck units are one of several systems of natural units, units of measurement that normalize certain fundamental physical constants to 1. Planck units normalize to 1 the constants shown in Table 1. These constants do not invoke the properties (mass, size, radius, or charge) of any elementary particle or macroscopic object, because the selection of such a particle or object would be necessarily arbitrary. This invariance to the properties of matter, and focus on the properties of free space, distinguishes Planck units from other systems of natural units. A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... In physics, natural units are physical units of measurement defined in terms of universal physical constants in such a manner that some chosen physical constants take on the numerical value of one when expressed in terms of a particular set of natural units. ... The former Weights and Measures office in Middlesex, England. ... In science, a physical constant is a physical quantity whose numerical value does not change. ... For other uses, see Mass (disambiguation). ... The elementary charge (symbol e or sometimes q) is the electric charge carried by a single proton, or equivalently, the negative of the electric charge carried by a single electron. ... For the novel, see The Elementary Particles. ... In physics, free space is a concept of electromagnetic theory, corresponding roughly to the vacuum, the baseline state of the electromagnetic field, or the replacement for the electromagnetic aether. ...

Planck units elegantly simplify many recurring algebraic expressions in theoretical physics, and are commonly employed in research on unified theories such as quantum gravity. The name "Planck units" honors Max Planck, who first proposed the base units in Table 2 (q_P excepted) in 1899. Nondimensionalization refers to the partial or full removal of units from a mathematical equation by a suitable substitution of variables. ... An expression is a combination of numbers, operators, grouping symbols (such as brackets and parentheses) and/or free variables and bound variables arranged in a meaningful way which can be evaluated. ... Quantum gravity is the field of theoretical physics attempting to unify quantum mechanics, which describes three of the fundamental forces of nature, with general relativity, the theory of the fourth fundamental force: gravity. ... “Planck” redirects here. ...

Table 1

L, M,T, Q, and Θ are explained in Table 2. For other uses of this word, see Length (disambiguation). ... For other uses, see Mass (disambiguation). ... This article is about the concept of time. ... This box:      Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. ... For other uses, see Temperature (disambiguation). ...

Constant	Symbol	Dimension	Related Theories
Speed of light in vacuum Dimensional analysis is a conceptual tool often applied in physics, chemistry, and engineering to understand physical situations involving a mix of different kinds of physical quantities. ... The speed of light in a vacuum is an important physical constant denoted by the letter c for constant or the Latin word celeritas meaning swiftness.[1] It is the speed of all electromagnetic radiation, including visible light, in a vacuum. ... Look up Vacuum in Wiktionary, the free dictionary. ...		L T −1	Special relativity Electromagnetism For a less technical and generally accessible introduction to the topic, see Introduction to special relativity. ... Electromagnetism is the physics of the electromagnetic field: a field which exerts a force on particles that possess the property of electric charge, and is in turn affected by the presence and motion of those particles. ...
Gravitation		L3 M−1 T −2	General relativity Newtonian gravitation According to the law of universal gravitation, the attractive force between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them. ... For a less technical and generally accessible introduction to the topic, see Introduction to general relativity. ... Isaac Newtons theory of universal gravitation (part of classical mechanics) states the following: Every single point mass attracts every other point mass by a force pointing along the line combining the two. ...
Dirac	where is Planck's constant	L2 M T −1	Quantum physics
Coulomb force	where is the permittivity of free space	L3 M T −2 Q−2	Electrostatics
Boltzmann		L2 M T −2 Θ−1	Thermodynamics Statistical mechanics Plancks constant, denoted h, is a physical constant that is used to describe the sizes of quanta. ... A commemoration plaque for Max Planck on his discovery of Plancks constant, in front of Humboldt University, Berlin. ... Fig. ... This box:      Coulombs torsion balance Coulombs law, developed in the 1780s by French physicist Charles Augustin de Coulomb, may be stated in scalar form as follows: The magnitude of the electrostatic force between two point electric charges is directly proportional to the product of the magnitudes of each... This article is in need of attention. ... Electrostatics (also known as static electricity) is the branch of physics that deals with the phenomena arising from what seem to be stationary electric charges. ... The Boltzmann constant (k or kB) is the physical constant relating temperature to energy. ... Thermodynamics (from the Greek θερμη, therme, meaning heat and δυναμις, dynamis, meaning power) is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics. ... Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ...

Base Planck units

Like all systems of measurement, Planck units feature "base units." In the SI (cgs) system, the base unit of length is the meter (centimeter). The Planck base unit of length is the Planck length, as defined in Table 2. Any other unit of length can be defined as some multiple (or fraction) of the Planck length, the meter, or the centimeter. The International System of Units (symbol: SI) (for the French phrase Système International dUnités) is the most widely used system of units. ... CGS is an acronym for centimetre-gram-second. ... The metre, or meter (symbol: m) is the SI base unit of length. ... This article is being considered for deletion in accordance with Wikipedias deletion policy. ... The Planck length, denoted by , is the unit of length approximately 1. ...

Like all systems of natural units, Planck units are derived by a form of dimensional analysis. Setting to unity the five fundamental constants in Table 1 defines the base units of length, mass, time, charge, and temperature shown in Table 2. These base units are derived from algebraic expressions, shown in the column "Expressions," combining the fundamental constants. These expressions are such that all dimensions cancel but one; that dimension has exponent 1. In physics, natural units are physical units of measurement defined in terms of universal physical constants in such a manner that some chosen physical constants take on the numerical value of one when expressed in terms of a particular set of natural units. ... Dimensional analysis is a conceptual tool often applied in physics, chemistry, and engineering to understand physical situations involving a mix of different kinds of physical quantities. ... For other uses of this word, see Length (disambiguation). ... For other uses, see Mass (disambiguation). ... This article is about the concept of time. ... This box:      Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. ... For other uses, see Temperature (disambiguation). ...

Table 2

Base Unit (Dimension) Dimensional analysis is a conceptual tool often applied in physics, chemistry, and engineering to understand physical situations involving a mix of different kinds of physical quantities. ...	Expressions	Approximate SI equivalent	Other equivalent
Length (L)		1.616252 × 10−35 m[1]
Mass (M)		2.17645 × 10−8 kg[2]	1.311 × 1019 u
Time (T)		5.39121 × 10−44 s[3]
Charge (Q)		1.8755459 × 10−18 C[4]	11.70624 e
Temperature (Θ)		1.41679 × 1032 K[5]

Each base unit is a function of c and . The same is true of G, charge excepted. ε₀ and k matter only for charge and temperature, respectively. For this reason, the expressions for the Planck charge and Planck temperature can be obtained by elementary algebra. Look up si, Si, SI in Wiktionary, the free dictionary. ... The Planck length, denoted by , is the unit of length approximately 1. ... (Redirected from 1 E-35 m) Categories: Orders of magnitude (length) ... This article is about the unit of length. ... The Planck mass is the natural unit of mass, denoted by mP. It is the mass for which the Schwarzschild radius is equal to the Compton length divided by Ï€. ≈ 1. ... (Redirected from 1 E 8 kg) Categories: Orders of magnitude (mass) ... Kg redirects here. ... The unified atomic mass unit (u), or dalton (Da), is a small unit of mass used to express atomic and molecular masses. ... In physics, the Planck time (tP), is the unit of time in the system of natural units known as Planck units. ... To help compare different orders of magnitudes this page lists times between 10-44s and 10-43s. ... This article is about the unit of time. ... In physics, the Planck charge is the unit of electric charge, denoted by , in the system of natural units known as Planck units. ... The coulomb (symbol: C) is the SI unit of electric charge. ... The elementary charge (symbol e or sometimes q) is the electric charge carried by a single proton, or equivalently, the negative of the electric charge carried by a single electron. ... The Planck temperature, named after German physicist Max Planck, is the natural unit of temperature, denoted by TP. The Planck units, in general, represent limits of quantum mechanics. ... (Redirected from 1 E32 K) To help compare different orders of magnitude this page lists temperatures above 1030 kelvins. ... For other uses, see Kelvin (disambiguation). ... In physics, the Planck charge is the unit of electric charge, denoted by , in the system of natural units known as Planck units. ... The Planck temperature, named after German physicist Max Planck, is the natural unit of temperature, denoted by TP. The Planck units, in general, represent limits of quantum mechanics. ...

Planck (1899) did not mention the Planck charge q_P, but the expression for it Table 2 is an elementary exercise in dimensional analysis, similar to how the definitions of the other base Planck units are obtained.^[6] q_P is closely related to the dimensionless fine structure constant, α, and e, the elementary charge, the charge that nature assigns to the electron and proton, the two stable charged particles: In physics, the Planck charge is the unit of electric charge, denoted by , in the system of natural units known as Planck units. ... Dimensional analysis is a conceptual tool often applied in physics, chemistry, and engineering to understand physical situations involving a mix of different kinds of physical quantities. ... The fine-structure constant or Sommerfeld fine-structure constant, usually denoted , is the fundamental physical constant characterizing the strength of the electromagnetic interaction. ... The elementary charge (symbol e or sometimes q) is the electric charge carried by a single proton, or equivalently, the negative of the electric charge carried by a single electron. ... The deepest visible-light image of the cosmos. ... For other uses, see Electron (disambiguation). ... For other uses, see Proton (disambiguation). ...

The first equality defines α; the second results from substituting the definition of q_P into that of α. Thus the numerical value of α is simply the square of the elementary charge expressed as a fraction of the Planck charge. Both e and α are well-understood and measured with considerable precision. In physics, the Planck charge is the unit of electric charge, denoted by , in the system of natural units known as Planck units. ...

From the preceding equation, the SI equivalent of the Planck charge can be calculated as: Look up si, Si, SI in Wiktionary, the free dictionary. ...

Because e is proportional to α^1/2, and the electromagnetic force between two charged particles is proportional to the product of the charge of each particle, the strength of the electromagnetic force relative to other fundamental forces is proportional to α. The factor of proportionality includes (q_P)². In physics, the electromagnetic force is the force that the electromagnetic field exerts on electrically charged particles. ...

Derivation by linear equations

One can think of the natural log of a base Planck unit as being a linear combination of the logs of the fundamental constants in Table 1. The coefficients of this linear combination are the exponents of the unlogged constants in the Table 2 definitions of the base units. The natural logarithm is the logarithm to the base e, where e is approximately equal to 2. ... In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ... In physics, fundamental physical constants are physical constants that are independent of systems of units and are in general dimensionless numbers. ... For other senses of this word, see coefficient (disambiguation). ... In mathematics, exponentiation is a process generalized from repeated multiplication, in much the same way that multiplication is a process generalized from repeated addition. ...

Let x be the vector (l_P, m_P, t_P) of base Planck units. b is the corresponding vector (c, G, ) of fundamental constants. Let A be a 3x3 matrix whose entries are the exponents of L, M, and T in the "Dimensions" column of Table 1. The rows (columns) of A correspond to the base Planck units (fundamental constants). Then the following system of linear equations relates the base Planck units and the fundamental constants: Look up matrix in Wiktionary, the free dictionary. ... In mathematics, exponentiation is a process generalized from repeated (or iterated) multiplication, in much the same way that multiplication is a process generalized from repeated addition. ... In mathematics and linear algebra, a system of linear equations is a set of linear equations such as A standard problem is to decide if any assignment of values for the unknowns can satisfy all three equations simultaneously, and to find such an assignment if it exists. ...

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Solve for the base Planck units in terms of the fundamental constants by solving for x in terms of A and b, or A^-1b, where A^-1 is the matrix inverse of A. The first row of A^-1 is the vector (-3/2, 1/2, 1/2). These are the exponents for c, G, and , respectively, in the expression for l_P in Table 2. The second (third) row of A^-1 contain the exponents for comparable expressions for m_P (t_P). In mathematics and especially linear algebra, an n-by-n matrix A is called invertible, non-singular or regular if there exists another n-by-n matrix B such that AB = BA = In, where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ... Look up vector in Wiktionary, the free dictionary. ...

Derived Planck units

Table 3 shows how other physical units, the derived Planck units, can be derived proximately from the base units and ultimately from the fundamental constants. c appears in the expression for every derived unit; the same holds for G, impedance excepted. appears only in the expressions for the first 5 of the 11 derived units. ε₀ matters only for voltage, current, and impedance. T_P and k do not appear in this Table.

Table 3

Derived Unit (Dimension) Dimensional analysis is a conceptual tool often applied in physics, chemistry, and engineering to understand physical situations involving a mix of different kinds of physical quantities. ...	Expression	Approximate SI equivalent
Angular frequency (T−1)		1.85487 × 1043 s−1
Area (L2)		2.61223 × 10-70 m2
Momentum (LMT−1)		6.52485 kg m/s
Density (ML−3)		5.15500 × 1096 kg/m3
Energy (L2MT−2)		1.9561 × 109 J
Power (L2MT−3)		3.62831 × 1052 W
Force (LMT−2)		1.21027 × 1044 N
Pressure (LM−1T−2)		4.63309 × 10113 Pa
Electric current (QT−1)		3.4789 × 1025 A
Voltage (L2MT−2Q−1)		1.04295 × 1027 V
Impedance (L2MT−1Q−2)		29.9792458 Ω

Look up si, Si, SI in Wiktionary, the free dictionary. ... The Planck angular frequency is the natural unit of angular frequency, denoted by ωP. 1. ... It has been suggested that this article or section be merged into Angular velocity. ... A square metre (US spelling: square meter) is by definition the area enclosed by a square with sides each 1 metre long. ... Planck Momentum is the unit of momentum, denoted by , in the system of natural units known as Planck units. ... Kg redirects here. ... Metre per second (U.S. spelling: meter per second) is an SI derived unit of both speed (scalar) and velocity (vector), defined by distance in metres divided by time in seconds. ... The Planck density is the natural unit of density, denoted by ρP. ρP = Planck mass / (Planck length)3 = ≈ 5. ... External links Conversion Calculator for Units of Density Category: ... Kilogram per cubic metre is the SI measure of density and is represented as kg/m³, where kg stands for kilogram and m³ stands for cubic metre. ... To help compare different orders of magnitude we list here energies between 109 joules (a gigajoule, symbol GJ) and 1010 joules. ... The joule (IPA: or ) (symbol: J) is the SI unit of energy. ... The Planck energy divided by the Planck time is the Planck power, equal to about 3. ... This page lists examples of the power in watts produced by various different sources of energy. ... For other uses, see Watt (disambiguation). ... Planck Force A derived Planck unit equated to the Planck Energy (also derived) divided by the Planck length. ... For other uses, see Newton (disambiguation). ... The Planck pressure is the natural unit of pressure, denoted by pP. 4. ... 1 At earth mean sea level. ... For other uses, see Pascal. ... 1 - conductors, Fp - Planck force, lp - Planck leght, Ip - Planck current. ... For other uses, see Ampere (disambiguation). ... The Planck voltage is the natural unit of voltage, denoted by VP. 1. ... Josephson junction array chip developed by NIST as a standard volt. ... The Planck impedance is the natural unit of impedance, denoted by ZP. 2. ... The ohm (symbol: Ω) is the SI unit of electric resistance. ...

Planck units simplify the key equations of physics

Table 4 shows how Planck units, by normalizing the five fundamental constants in Table 1 to unity, simplify many equations of physics and make them nondimensional. Nondimensionalization refers to the partial or full removal of units from a mathematical equation by a suitable substitution of variables. ...

Table 4

	Usual form	Nondimensionalized form
Newton's Law of universal gravitation
Schrödinger's equation
Equation relating particle energy to the radian frequency of the wave function
Einstein's mass/energy equation of special relativity
Einstein's field equation for general relativity
Thermal energy per particle per degree of freedom
Coulomb's law
Maxwell's equations	Nondimensionalization refers to the partial or full removal of units from a mathematical equation by a suitable substitution of variables. ... Gravity is a force of attraction that acts between bodies that have mass. ... In physics, the Schrödinger equation, proposed by the Austrian physicist Erwin Schrödinger in 1925, describes the time-dependence of quantum mechanical systems. ... For the novel, see The Elementary Particles. ... For the musical group, see Radian (band). ... A wave function is a mathematical tool that quantum mechanics uses to describe any physical system. ... Einstein redirects here. ... For other uses, see Mass (disambiguation). ... For a less technical and generally accessible introduction to the topic, see Introduction to special relativity. ... In physics, the Einstein field equation or Einstein equation is a differential equation in Einsteins theory of general relativity. ... For a less technical and generally accessible introduction to the topic, see Introduction to general relativity. ... In thermal physics, thermal energy is the energy portion of a system that increases with its temperature. ... Degrees of freedom is a general term used in explaining dependence on parameters, and implying the possibility of counting the number of those parameters. ... This box:      Coulombs torsion balance Coulombs law, developed in the 1780s by French physicist Charles Augustin de Coulomb, may be stated in scalar form as follows: The magnitude of the electrostatic force between two point electric charges is directly proportional to the product of the magnitudes of each... For thermodynamic relations, see Maxwell relations. ...

Discussion

Natural units began in 1881, when George Johnstone Stoney derived units of length, time, and mass, now named Stoney units in his honor, by normalizing G, c, and the electron charge e to 1. (Stoney was also the first to hypothesize that electric charge is quantized and hence to see the fundamental character of e.) The constants now named after Planck and Boltzmann were then unknown. Replacing in the expression defining a Planck unit with e²/c yields the corresponding Stoney unit. Since with α dimensionless, and Planck units are a function of , the SI numerical equivalents of a Stoney unit and its Planck analog differ by one order of magnitude, the factor .^[7] In physics, natural units are physical units of measurement defined in terms of universal physical constants in such a manner that some chosen physical constants take on the numerical value of one when expressed in terms of a particular set of natural units. ... George Johnstone Stoney (February 15, 1826 – July 5, 1911) was an Irish physicist. ... The elementary charge (symbol e or sometimes q) is the electric charge carried by a single proton, or equivalently, the negative of the electric charge carried by a single electron. ... The fine-structure constant or Sommerfeld fine-structure constant, usually denoted , is the fundamental physical constant characterizing the strength of the electromagnetic interaction. ... An order of magnitude is the class of scale or magnitude of any amount, where each class contains values of a fixed ratio to the class preceding it. ...

Natural units can help physicists reframe questions. An example of such reframing is the following passage by Frank Wilczek: Frank Wilczek (born May 15, 1951) is a Nobel prize winning American physicist. ...

...We see that the question is not, "Why is gravity so feeble?" but rather, "Why is the proton's mass so small?" For in Natural (Planck) Units, the strength of gravity simply is what it is, a primary quantity, while the proton's mass is the tiny number [1/(13 quintillion)]...

– June 2001 Physics Today

Max Planck first set out the base units (q_P excepted) later named in his honor, in a paper presented to the Prussian Academy of Sciences in May 1899.^[8] That paper also includes the first appearance of a constant named b, and later called h and named after him. The paper gave numerical values for the base units, in terms of the metric system of his day, that were remarkably close to those in Table 2. We are not sure just how Planck came to discover these units because his paper gave no algebraic details. But he did explain why he valued these units as follows: “Planck” redirects here. ... A commemoration plaque for Max Planck on his discovery of Plancks constant, in front of Humboldt University, Berlin. ...

...ihre Bedeutung für alle Zeiten und für alle, auch außerirdische und außermenschliche Kulturen notwendig behalten und welche daher als »natürliche Maßeinheiten« bezeichnet werden können...

...These necessarily retain their meaning for all times and for all civilizations, even extraterrestrial and non-human ones, and can therefore be designated as "natural units"...

Physicists sometimes humorously refer to Planck units as "God's units," as Planck units are free of arbitrary anthropocentricity. While the SI meter and second are associated mainly for historical reasons, the corresponding Planck length and Planck time are conceptually linked at a fundamental physical level. Anthropocentrism (Greek άνθρωπος, anthropos, human, κέντρον, kentron, center), or the human-centered principle, refers to the idea that humanity must always remain the central concern for humans. ... The International System of Units (symbol: SI) (for the French phrase Système International dUnités) is the most widely used system of units. ... The metre, or meter (symbol: m) is the SI base unit of length. ... This article is about the unit of time. ... The Planck length, denoted by , is the unit of length approximately 1. ... In physics, the Planck time (tP), is the unit of time in the system of natural units known as Planck units. ...

Some Planck units are relatively easy to grasp. One unit of:

• Planck mass is about 20 micrograms;
• Planck momentum is based on a mass of 6.5kg;
• Planck energy is about 500Kwh;
• Planck charge is only one order of magnitude greater than the elementary charge;
• Planck impedance is very nearly 30 ohms.

But the other Planck units are many orders of magnitude too large or too small to be of any empirical and practical use, so that Planck units as a system are relevant only to theoretical physics. In fact, 1 Planck unit is often the largest or smallest value of a physical quantity that makes sense given the current state of physical theory. For instance: The Planck mass is the natural unit of mass, denoted by mP. It is the mass for which the Schwarzschild radius is equal to the Compton length divided by Ï€. ≈ 1. ... In the metric system, a microgram is 1/1,000,000 of a gram, or 1/1000 of a milligram, is one of the smallest units of weight/mass commonly used. ... Planck Momentum is the unit of momentum, denoted by , in the system of natural units known as Planck units. ... The Planck energy is the natural unit of energy, denoted by EP. 1. ... The kilowatt-hour (symbol: kW·h) is a unit for measuring energy. ... In physics, the Planck charge is the unit of electric charge, denoted by , in the system of natural units known as Planck units. ... The elementary charge (symbol e or sometimes q) is the electric charge carried by a single proton, or equivalently, the negative of the electric charge carried by a single electron. ... The Planck impedance is the natural unit of impedance, denoted by ZP. 2. ... The ohm is the SI derived unit of electrical resistance (derived from the ampere and the watt). ... An order of magnitude is the class of scale or magnitude of any amount, where each class contains values of a fixed ratio to the class preceding it. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ...

• A Planck velocity of 1 is the speed of light in a vacuum, the maximum possible velocity given special relativity;
• At a Planck temperature of 1, all symmetries broken since the early Big Bang are restored, and the four fundamental forces of contemporary physical theory become one force;
• Our understanding of the Big Bang begins when the universe attained an age and size of about 1 Planck time and 1 Planck length, at which time its Planck temperature was about 1 and quantum theory as presently understood begins to hold. Understanding the universe when it was less than 1 Planck time old requires a theory of quantum gravity, incorporating quantum effects into general relativity. Such a theory does not yet exist.

This article is about velocity in physics. ... The speed of light in a vacuum is an important physical constant denoted by the letter c for constant or the Latin word celeritas meaning swiftness.[1] It is the speed of all electromagnetic radiation, including visible light, in a vacuum. ... For a less technical and generally accessible introduction to the topic, see Introduction to special relativity. ... The Planck temperature, named after German physicist Max Planck, is the natural unit of temperature, denoted by TP. The Planck units, in general, represent limits of quantum mechanics. ... Promotional picture Symmetry Breaking is a rock band from Northern New Jersey, in the United States. ... For other uses, see Big Bang (disambiguation). ... A fundamental interaction or fundamental force is a mechanism by which particles interact with each other, and which cannot be explained in terms of another interaction. ... For other uses, see Big Bang (disambiguation). ... In physics, the Planck time (tP), is the unit of time in the system of natural units known as Planck units. ... The Planck length, denoted by , is the unit of length approximately 1. ... The Planck temperature, named after German physicist Max Planck, is the natural unit of temperature, denoted by TP. The Planck units, in general, represent limits of quantum mechanics. ... For a generally accessible and less technical introduction to the topic, see Introduction to quantum mechanics. ... In physics, the Planck time (tP), is the unit of time in the system of natural units known as Planck units. ... Quantum gravity is the field of theoretical physics attempting to unify quantum mechanics, which describes three of the fundamental forces of nature, with general relativity, the theory of the fourth fundamental force: gravity. ... For a generally accessible and less technical introduction to the topic, see Introduction to quantum mechanics. ... For a less technical and generally accessible introduction to the topic, see Introduction to general relativity. ...

Error propagation

c appears in every Planck unit defined in Tables 2 and 3. However, the speed of light in SI units is no longer subject to measurement error, because the SI base unit of length, the meter, is now defined as some stipulated fraction of the distance light travels in 1 second. Hence the value of c is now exact by definition, and c contributes no uncertainty to the SI equivalents of the Planck units. The same is true of ε₀.^[9] The numerical value of has been determined experimentally to about 1 part in 50,000,000.^[10] The speed of light in a vacuum is an important physical constant denoted by the letter c for constant or the Latin word celeritas meaning swiftness.[1] It is the speed of all electromagnetic radiation, including visible light, in a vacuum. ... Look up si, Si, SI in Wiktionary, the free dictionary. ... Look up si, Si, SI in Wiktionary, the free dictionary. ... The metre, or meter (symbol: m) is the SI base unit of length. ... Plancks constant, denoted h, is a physical constant that is used to describe the sizes of quanta. ...

Meanwhile, the numerical value of G has been determined experimentally to no better than 1 part in 10000.^[11] G also appears in the definition of almost every Planck unit in Tables 2 and 3. Hence the uncertainty in the values of the Table 2 and 3 SI equivalents of the Planck units derives almost entirely from uncertainty in the value of G. The propagation of the error in G is a function of the exponent of G in the algebraic expression for a unit. Since that exponent is ±¹⁄₂ for every base unit, the uncertainty of each base unit is about one half that of G. This is indeed the case; according to CODATA, the experimental values of the SI equivalents of the base Planck units are known to about 1 part in 20,000. According to the law of universal gravitation, the attractive force between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them. ... According to the law of universal gravitation, the attractive force between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them. ...

Measuring the universe in Planck units

Table 5 sets out certain basic facts about the present-day observable universe, when measured in orders of magnitude of Planck units. See universe for a general discussion of the universe. ... An order of magnitude is the class of scale or magnitude of any amount, where each class contains values of a fixed ratio to the class preceding it. ...

Table 5

Table 5 makes precise how the present-day universe is very cold. It is also essentially empty, because its mass and diameter have the same order of magnitude, in which case (assuming the universe is spherical) its density is inversely related to the square of its diameter. In physics, the Planck time (tP), is the unit of time in the system of natural units known as Planck units. ... This article is about the unit of time. ... In physics, the Planck time (tP), is the unit of time in the system of natural units known as Planck units. ... This box:      This article is about scientific estimates of the age of the universe. ... The Planck length, denoted by , is the unit of length approximately 1. ... The Planck length, denoted by , is the unit of length approximately 1. ... The Planck length, denoted by , is the unit of length approximately 1. ... See universe for a general discussion of the universe. ... The Planck mass is the natural unit of mass, denoted by mP. It is the mass for which the Schwarzschild radius is equal to the Compton length divided by Ï€. ≈ 1. ... See universe for a general discussion of the universe. ... For other uses, see Proton (disambiguation). ... The Planck mass is the natural unit of mass, denoted by mP. It is the mass for which the Schwarzschild radius is equal to the Compton length divided by Ï€. ≈ 1. ... Kg redirects here. ... See universe for a general discussion of the universe. ... The Planck temperature, named after German physicist Max Planck, is the natural unit of temperature, denoted by TP. The Planck units, in general, represent limits of quantum mechanics. ... When any patch of the sky is observed where no individual sources can be discerned, and the effects of interplanetary dust, and interstellar matter are taken into account, there is still radiation. ... The kelvin (symbol: K) is the SI unit of temperature, and is one of the seven SI base units. ... For other uses, see Temperature (disambiguation). ...

The present temperature of the universe is that of the cosmic background radiation and so is known precisely. Its age and size are moderately well measured. But the mass of the universe (and hence its density) are more of an educated guess. The standard approach to estimating this mass assumes that: When any patch of the sky is observed where no individual sources can be discerned, and the effects of interplanetary dust, and interstellar matter are taken into account, there is still radiation. ...

• The stable leptons (the electrons and neutrinos) contribute negligeable mass to the universe. This is not controversial;
• All mass consists of stars, black holes, and the interstellar medium, assumed to consist of hydrogen. Hence the mass of the universe is entirely composed of stable hadrons, namely protons and bound neutrons. This rules out having dark matter and dark energy contribute to the mass of the universe.

While the value of 10⁸⁰ for the number of protons in the universe (sometimes called the Eddington number) is fairly well-established, it is not a measurement. If this number were in fact 10⁸¹, then every value in the middle column would be an integer power of 10³¹. (Temperature violates this by only 1 order of magnitude.) The comparable values in Barrow (2002: 119-20) are integer powers of 10³⁰, a fact presented without explanation. These are large number coincidences of the sort first proposed by Paul Dirac, and discussed in Barrow (2002: chpt. 6) and Barrow and Tipler (1986: chpt. 4). In physics, a lepton is a particle with spin-1/2 (a fermion) that does not experience the strong interaction (that is, the strong nuclear force). ... For other uses, see Electron (disambiguation). ... For other uses, see Neutrino (disambiguation). ... This article is about the astronomical object. ... For other uses, see Black hole (disambiguation). ... The interstellar medium (or ISM) is the name astronomers give to the tenuous gas and dust that pervade interstellar space. ... This article is about the chemistry of hydrogen. ... A hadron, in particle physics, is a subatomic particle which experiences the nuclear force. ... For alternative meanings see proton (disambiguation). ... This article or section does not adequately cite its references or sources. ... For other uses, see Dark matter (disambiguation). ... In physical cosmology, dark energy is a hypothetical form of energy that permeates all of space and tends to increase the rate of expansion of the universe. ... See universe for a general discussion of the universe. ... In 1938, the British astronomer Arthur Eddington hit on the idea that the fine structure constant α, which had been measured at approximately 1/136, should be exactly 1/136. ... An order of magnitude is the class of scale or magnitude of any amount, where each class contains values of a fixed ratio to the class preceding it. ... The Dirac large numbers hypothesis refers to an observation made by Paul Dirac in 1937 relating ratios of size scales in the universe to that of force scales. ... Paul Adrien Maurice Dirac, OM, FRS (IPA: [dɪræk]) (August 8, 1902 – October 20, 1984) was a British theoretical physicist and a founder of the field of quantum physics. ...

Alternative normalizations

As already stated, Planck units are derived by "normalizing" the numerical values of certain fundamental constants to 1. These normalizations are neither the only ones possible, nor necessarily the best. Moreover, the choice of what constants to normalize is not evident, and the values of the Planck units are sensitive to this choice.

The factor n4π, n=1,2, or 4, is ubiquitous in physical theory because it appears in the formulas for the surface area and volume of a 3-dimensional sphere. If space had more than 3 dimensions, the factor corresponding to 4π would be much larger. Gravitational and electrostatic fields are sphere-like in that their strengths vary with distance but not direction (Barrow 2002: 214-15). In any event, a fundamental choice that has to be made when designing a system of natural units is which, if any, instances of n4π appearing in the equations of physics are to be eliminated via normalization. Theoretical physics attempts to understand the world by making a model of reality, used for rationalizing, explaining, predicting physical phenomena through a physical theory. There are three types of theories in physics; mainstream theories, proposed theories and fringe theories. ... For other uses, see Sphere (disambiguation). ... A gravitational field is a model used within physics to explain how gravity exists in the universe. ... In physics, an electric field or E-field is an effect produced by an electric charge that exerts a force on charged objects in its vicinity. ...

Normalizing Boltzmann's constant k to 2. This would remove the factor of 2 in the nondimensionalized equation for the thermal energy per particle per degree of freedom, and would not affect the value of any base or derived unit other than the Planck temperature. The Boltzmann constant (k or kB) is the physical constant relating temperature to energy. ... Nondimensionalization refers to the partial or full removal of units from a mathematical equation by a suitable substitution of variables. ... In thermal physics, thermal energy is the energy portion of a system that increases with its temperature. ... Degrees of freedom is a general term used in explaining dependence on parameters, and implying the possibility of counting the number of those parameters. ... The Planck temperature, named after German physicist Max Planck, is the natural unit of temperature, denoted by TP. The Planck units, in general, represent limits of quantum mechanics. ...

Eliminating 4π from the equations for electromagnetism. Planck's choices of what to normalize were also a consequence of the state of physical theory in 1899. When he introduced the units now named after him, the understanding of electromagnetism was not what is today, so that Coulomb's law was seen as more fundamental than Maxwell's equations. Hence Planck normalized to 1 the Coulomb force constant (4πε₀)⁻¹ (as does the cgs system of units). This sets the Planck impedance, Z_P equal to Z₀/4π, where Z₀ is the characteristic impedance of free space. Normalizing the permittivity of free space ε₀ to 1 not only makes Z_P equal to Z₀ but also eliminates 4π from Maxwell's equations. On the other hand, the nondimensionalized form of Coulomb's law would now include a factor of (4π)⁻¹. Electromagnetism is the physics of the electromagnetic field: a field which exerts a force on particles that possess the property of electric charge, and is in turn affected by the presence and motion of those particles. ... Electromagnetism is the physics of the electromagnetic field: a field which exerts a force on particles that possess the property of electric charge, and is in turn affected by the presence and motion of those particles. ... This box:      Coulombs torsion balance Coulombs law, developed in the 1780s by French physicist Charles Augustin de Coulomb, may be stated in scalar form as follows: The magnitude of the electrostatic force between two point electric charges is directly proportional to the product of the magnitudes of each... For thermodynamic relations, see Maxwell relations. ... In physics, Coulombs law is an inverse-square law indicating the magnitude and direction of electrostatic force that one stationary, electrically charged object of small dimensions (ideally, a point source) exerts on another. ... This article or section is in need of attention from an expert on the subject. ... The Planck impedance is the natural unit of impedance, denoted by ZP. 2. ... The characteristic impedance of vacuum or characteristic impedance of free space (Z0) is a physical constant, the characteristic impedance of electromagnetic radiation in vacuum, defined by: where: = magnetic constant = electric constant = speed of light In SI units, the value is exactly expressed by: = 1. ... This article is in need of attention. ... For thermodynamic relations, see Maxwell relations. ... Nondimensionalization refers to the partial or full removal of units from a mathematical equation by a suitable substitution of variables. ... This box:      Coulombs torsion balance Coulombs law, developed in the 1780s by French physicist Charles Augustin de Coulomb, may be stated in scalar form as follows: The magnitude of the electrostatic force between two point electric charges is directly proportional to the product of the magnitudes of each...

Eliminating n4π from the equations for general relativity and cosmology (see geometrized unit system). In 1899, general relativity lay some years in the future, so that Newton's law was still seen as fundamental, rather than as a convenient approximation holding for "small" velocities and distances. Hence Planck normalized to 1 the G in Newton's law of universal gravitation. In theories emerging after 1899, G is nearly always multiplied by n4π, n=1,2, or 4: For a less technical and generally accessible introduction to the topic, see Introduction to general relativity. ... Cosmology, from the Greek: κοσμολογία (cosmologia, κόσμος (cosmos) order + λογια (logia) discourse) is the study of the Universe in its totality, and by extension, humanitys place in it. ... In physics, especially in the general theory of relativity, geometrized units or geometric units constitute a physical unit system in which all physical quantities are identified with geometric quantities such as areas, lengths, dimensionless numbers, path curvatures, or sectional curvatures. ... For a less technical and generally accessible introduction to the topic, see Introduction to general relativity. ... It has been suggested that this article or section be merged into Gravity. ... According to the law of universal gravitation, the attractive force between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them. ... It has been suggested that this article or section be merged into Gravity. ...

• 4πG appears in the:
  • Gravitational form of Gauss's Law, Φ_g = − 4πGM;
  • Characteristic impedance of gravitational radiation in free space, Z₀ = 4πG/c. The c in the denominator stems from the general relativity result that gravitational radiation propagates at the same velocity as electromagnetic radiation;
  • Gravitoelectromagnetic (GEM) equations, which hold in weak gravitational fields or reasonably flat space-time. These equations have the same form as Maxwell's equations (and the Lorentz force equation) of electromagnetism, with mass(mass density) replacing charge (charge density), and (4πG)⁻¹ replacing the permittivity ε₀.
• 8πG appears in the Einstein field equations, Einstein-Hilbert action, Friedmann equations, and the Poisson equation for gravitation. Planck units modified so that 8πG=1 are known as reduced Planck units because the factor (8π)^1/2 divides the Planck mass.
• 16πG. Normalizing this to 1 eliminates the constant from the Einstein-Hilbert action. The Einstein field equations with cosmological constant Λ becomes R_ab − Λg_ab = (Rg_ab − T_ab) / 2.

Hence a substantial body of physical theory discovered since Planck (1899) suggests normalizing to 1 not G but n4πG, n=1,2, or 4. However, doing so would introduce a factor of (n4π)⁻¹ into the nondimensionalized law of universal gravitation. In physics and mathematical analysis, Gausss law is the electrostatic application of the generalized Gausss theorem giving the equivalence relation between any flux, e. ... The characteristic impedance of a uniform transmission line is the ratio of the amplitudes of a single pair of voltage and current waves propagating along the line in the absence of reflections. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... In physics, free space is a concept of electromagnetic theory, corresponding roughly to the vacuum, the baseline state of the electromagnetic field, or the replacement for the electromagnetic aether. ... For a less technical and generally accessible introduction to the topic, see Introduction to general relativity. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... This box:      Electromagnetic (EM) radiation is a self-propagating wave in space with electric and magnetic components. ... This article is in need of attention from an expert on the subject. ... A gravitational field is a model used within physics to explain how gravity exists in the universe. ... In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ... For thermodynamic relations, see Maxwell relations. ... Lorentz force. ... Electromagnetism is the physics of the electromagnetic field: a field which exerts a force on particles that possess the property of electric charge, and is in turn affected by the presence and motion of those particles. ... Density (symbol: ρ - Greek: rho) is a measure of mass per unit of volume. ... Charge density is the amount of electric charge per unit volume. ... This article is in need of attention. ... The Einstein field equations (EFE) or Einsteins equations are a set of ten equations in Einsteins theory of general relativity in which the fundamental force of gravitation is described as a curved spacetime caused by matter and energy. ... In general relativity, Einsteins field equations can be derived from an action principle starting from the Einstein-Hilbert action: where g is the (pseudo)Riemannian metric, R is the Ricci scalar, n is the number of spacetime dimensions and k is a constant which depends on the units chosen... The Friedmann equations relate various cosmological parameters within the context of general relativity. ... Poissons equation is the partial differential equation: Or alternately: or i. ... The Planck mass is the natural unit of mass, denoted by mP. It is the mass for which the Schwarzschild radius is equal to the Compton length divided by Ï€. ≈ 1. ... In general relativity, Einsteins field equations can be derived from an action principle starting from the Einstein-Hilbert action: where g is the (pseudo)Riemannian metric, R is the Ricci scalar, n is the number of spacetime dimensions and k is a constant which depends on the units chosen... The Einstein field equations (EFE) or Einsteins equations are a set of ten equations in Einsteins theory of general relativity in which the fundamental force of gravitation is described as a curved spacetime caused by matter and energy. ... In physical cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: Λ) was proposed by Albert Einstein as a modification of his original theory of general relativity to achieve a stationary universe. ... Nondimensionalization refers to the partial or full removal of units from a mathematical equation by a suitable substitution of variables. ... It has been suggested that this article or section be merged into Gravity. ...

Planck units and the invariant scaling of nature

Dirac (1937), and others after him, have conjectured that some physical "constants" might actually change over time, a proposition that introduces many difficult questions such as:

• How would such a change make a noticeable operational difference in physical measurement or, more basically, our perception of reality?
• If some physical constant had changed, would we even notice it?
• How would physical reality be different?
• Which changed constants would result in a meaningful and measurable difference?

John Barrow has spoken to these questions as follows: This article is about the English statesman Sir John Barrow. ...

[An] important lesson we learn from the way that pure numbers like α define the world is what it really means for worlds to be different. The pure number we call the fine structure constant and denote by α is a combination of the electron charge, e, the speed of light, c, and Planck's constant, h. At first we might be tempted to think that a world in which the speed of light was slower would be a different world. But this would be a mistake. If c, h, and e were all changed so that the values they have in metric (or any other) units were different when we looked them up in our tables of physical constants, but the value of α remained the same, this new world would be observationally indistinguishable from our world. The only thing that counts in the definition of worlds are the values of the dimensionless constants of Nature. If all masses were doubled in value [including the Planck mass m_P] you cannot tell because all the pure numbers defined by the ratios of any pair of masses are unchanged.

– Barrow 2002

When measuring a length with a ruler or tape measure, one is actually counting tick marks on a given standard, i.e., measuring the length relative to that given standard; the result is a dimensionless value. It is no different for physical experiments, as all physical quantities are measured relative to some other like-dimensioned values. If all physical quantities (masses and other properties of particles) were expressed in terms of Planck units, those quantities would be dimensionless numbers (mass divided by the Planck mass, length divided by the Planck length, etc.) and the only quantities we would measure when observing nature or conducting experiments would be dimensionless numbers. See Duff (2004) and section III.5 (by Duff alone) of Duff, Okun, and Veneziano (2002). Gabriele Veneziano (b. ...

We can notice a difference if some dimensionless physical quantity such as α or the proton/electron mass ratio changes; either change would alter atomic structures. But if all dimensionless physical quantities remained constant (this includes all possible ratios of identically dimensioned physical quantities), we could not tell if a dimensionful quantity, such as the speed of light, c, had changed. And, indeed, the Tompkins concept becomes meaningless in our existence if a dimensional quantity such as c has changed, even drastically. The variable speed of light (VSL) concept states that the speed of light in vacuum, usually denoted by c, may not be constant, for some reason. ...

If the speed of light c, were somehow suddenly cut in half and changed to ^c⁄₂, (but with all dimensionless physical quantities continuing to remain constant), then the Planck length would increase by a factor of √(8) from the point-of-view of some unaffected "god-like" observer on the outside. But then the size of atoms (approximately the Bohr radius) are related to the Planck length by an unchanging dimensionless constant: The Planck length, denoted by , is the unit of length approximately 1. ... In the Bohr model of the structure of an atom, put forward by Niels Bohr in 1913, electrons orbit a central nucleus. ...

Atoms would then be bigger (in one dimension) by √(8), each of us would be taller by √(8), and so would our meter sticks be taller (and wider and thicker) by a factor of √(8) and we would not know the difference. Our perception of distance and lengths relative to the Planck length is logically an unchanging dimensionless constant.

Moreover, our clocks would tick slower by a factor of √(32) (from the point-of-view of this unaffected "god-like" observer) because the Planck time has increased by √(32) but we would not know the difference. (Our perception of durations of time relative to the Planck time is, axiomatically, an unchanging dimensionless constant.) This hypothetical god-like outside observer might observe that light now travels at half the speed that it used to (as well as all other observed velocities) but it would still travel 299792458 of our new meters in the time elapsed by one of our new seconds (^c⁄₂ ^√(32)⁄_√(8) continues to equal 299792458 m/s). We would not notice any difference. The speed of light in a vacuum is an important physical constant denoted by the letter c for constant or the Latin word celeritas meaning swiftness.[1] It is the speed of all electromagnetic radiation, including visible light, in a vacuum. ...

This contradicts what George Gamow wrote in his book Mr. Tompkins; where he suggested that if a dimension-dependent universal constant such as c changed, we would easily notice the difference. The disagreement stems from the ambiguity in the phrase "changing a physical constant"; what would happen depends on whether we hold constant all other (1)  dimensionless constants, or (2)  dimension-dependent constants. (2) is a somewhat confusing alternative, since most of our units of measurement are defined in relation to the outcomes of physical experiments, and the experimental results depend on the constants. (The only exception is the kilogram.) Gamow does not address this subtlety; the thought experiments he conducts in his popular works tacitly assume that (2) defines a "changing physical constant." George Gamow (pronounced GAM-off) (March 4, 1904 – August 19, 1968) , born Georgiy Antonovich Gamov (Георгий Антонович Гамов) was a Ukrainian born physicist and cosmologist. ... The eponymous character of Mr. ... Kg redirects here. ...

Footnotes

References

• John D. Barrow, 2002. The Constants of Nature; From Alpha to Omega - The Numbers that Encode the Deepest Secrets of the Universe. Pantheon Books. ISBN 0-375-42221-8. Easier.
• -------- and Frank J. Tipler, 1986. The Anthropic Cosmological Principle. Oxford University Press. Harder.
• Michael Duff, 2004, "Comment on time-variation of fundamental constants."
• --------, L. B. Okun, and Gabriele Veneziano, 2002, "Trialogue on the number of fundamental constants," Journal of High Energy Physics.
• Max Planck, 1899, "Über Irreversible Strahlungsvorgänge," ("On irreversible radiative processes") Sitzungsbericht Deutsche Akad. Wiss. Berlin, Math-Phys Tech. Kl 5: 440-80. Pp. 478-80 contain the first appearance of the Planck base units other than the Planck charge, and of Planck's constant, which Planck denoted by b. a and f in this paper correspond to k and G in this entry.
• Roger Penrose, 2004. The Road to Reality. Alfred A. Knopf. Section 31.1.
• Tomilin, K. A., 1999, "Natural Systems of Units: To the Centenary Anniversary of the Planck System," 287-96.

John David Barrow FRS (born November 29, 1952, London) is an English cosmologist, theoretical physicist, and mathematician. ... Frank J. Tipler (born in 1947 in Andalusia, Alabama) is a professor of mathematical physics at Tulane University in New Orleans, Louisiana. ... In physics and cosmology, the anthropic principle states that we should take into account the constraints that our existence as observers imposes on the sort of universe that we could observe. ... Sir Michael Duff, the bon vivant and society figure, was the son of Sir Robin Duff, 2nd Bt, of Vaynol, and his wife Lady Juliet Lowther, only child of the 4th Earl of Lonsdale and his wife Lady Gwladys Herbert (later Marchioness of Ripon). ... Gabriele Veneziano (b. ... “Planck” redirects here. ... In physics, the Planck charge is the unit of electric charge, denoted by , in the system of natural units known as Planck units. ... A commemoration plaque for Max Planck on his discovery of Plancks constant, in front of Humboldt University, Berlin. ... The Boltzmann constant (k or kB) is the physical constant relating temperature to energy. ... According to the law of universal gravitation, the attractive force between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them. ... Sir Roger Penrose, OM, FRS (born 8 August 1931) is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College. ... The Road to Reality is a book by the British mathematical physicist Roger Penrose, published in 2004. ...

External links

• Value of the fundamental constants, including the Planck base units, as reported by the National Institute of Standards and Technology (NIST).
• Sections C-E of collection of resources bear on Planck units. Good discussion of why 8πG should be normalized to 1 when doing general relativity and quantum gravity. Many links.